Combinatorial theory of Macdonald polynomials I: Proof of Haglund's formula

Abstract

Haglund recently proposed a combinatorial interpretation of the modified Macdonald polynomials H̃ μ . We give a combinatorial proof of this conjecture, which establishes the existence and integrality of H̃ μ . As corollaries, we obtain the cocharge formula of Lascoux and Schützenberger for Hall–Littlewood polynomials, a formula of Sahi and Knop for Jack's symmetric functions, a generalization of this result to the integral Macdonald polynomials J μ , a formula for H̃ μ in terms of Lascoux–Leclerc–Thibon polynomials, and combinatorial expressions for the Kostka–Macdonald coefficients K̃ λ,μ when μ is a two-column shape.